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A242966 Composite numbers whose anti-divisors are all primes.

%I A242966 #26 Mar 31 2015 03:59:00
%S A242966 4,8,16,64,1024,4096,65536,262144,4194304,1073741824,
%T A242966 1152921504606846976,1267650600228229401496703205376,
%U A242966 85070591730234615865843651857942052864,93536104789177786765035829293842113257979682750464
%N A242966 Composite numbers whose anti-divisors are all primes.
%C A242966 It appears they are all powers of 2.
%C A242966 Subset of A242965.
%C A242966 a(n) must be 2^k. - _Hiroaki Yamanouchi_, Mar 17 2015
%C A242966 The exponents are 2, 3, 4, 6, 10, 12, 16, 18, 22, 30, 60, 100, 126, 166, 198, ... - _Michel Marcus_, Mar 18 2015
%H A242966 Hiroaki Yamanouchi, <a href="/A242966/b242966.txt">Table of n, a(n) for n = 1..15</a>
%e A242966 The anti-divisors of 1024 are all primes: 3, 23, 89, 683.
%e A242966 The same for 65536: 3, 43691.
%p A242966 P := proc(q) local k,ok,n; for n from 3 to q do if not isprime(n)
%p A242966 then ok:=1; for k from 2 to n-1 do if abs((n mod k)-k/2)<1
%p A242966 then if not isprime(k) then ok:=0; break; fi; fi; od;
%p A242966 if ok=1 then print(n); fi; fi; od; end: P(10^100);
%t A242966 antiDivisors[n_] := Cases[Range[2, n - 1], _?(Abs[Mod[n, #] - #/2] < 1 &)]; Select[2^Range[2, 20], AllTrue[antiDivisors@ #, PrimeQ] &] (* _Michael De Vlieger_, Mar 18 2015 *)
%o A242966 (Python)
%o A242966 from sympy import isprime, divisors
%o A242966 A242966 = [n for n in range(3,10**5) if not isprime(n) and list(filter(lambda x: not isprime(x), [2*d for d in divisors(n) if n > 2*d and n % (2*d)] + [d for d in divisors(2*n-1) if n > d >=2 and n % d] + [d for d in divisors(2*n+1) if n > d >=2 and n % d])) == []]
%o A242966 # _Chai Wah Wu_, Aug 13 2014
%Y A242966 Cf. A066272, A242965.
%K A242966 nonn
%O A242966 1,1
%A A242966 _Paolo P. Lava_, May 28 2014
%E A242966 a(11)-a(14) from _Hiroaki Yamanouchi_, Mar 17 2015